We develop a queueing network model to analyze capacity planning and screening guideline decisions for a social planner who minimizes the long-run average total of detection delay cost and capacity cost. Patients spend a random amount of time at home before attempting to schedule a screen for which they must wait according to the queueing dynamics at the endoscopy suite. Between any two screens, patients may either die from other causes or develop cancer. If cancer is developed, it may be detected via a screen or via other methods. We solve for the optimal capacity decision when the average time patients spend at home is exogenous, and we perform sensitivity analysis on the optimal solution. When the capacity is xed and the provider optimizes the average time at home, the function is not convex in the general case so we solve a special case and perform sensitivity analysis. We investigate the general case via numerical study. When we allow the provider to jointly optimize both the capacity decision and the average time at home, we consider a special case for our analytical results because it is dicult to show unimodality and to solve the model in the general case. We use a numerical
study to generate real-world estimates of our results from public health data and to further investigate the model under general assumptions.
A primary objective and contribution for this research, beyond developing a model to op- timize and guide capacity planning decisions for colorectal cancer screening, is capturing and understanding the tradeo between screening guidelines and screening capacity. If screening guidelines are developed using medical decision making which only takes into account the utility and health state tradeos for a single individual, they may suggest policies which are unsus- tainable or sub-optimal given the system capacity or cost of adding additional capacity. If we assume the average time at home to be exogenous, e.g. due to well-established guidelines, we nd that increasing the average time at home causes the optimal capacity to increase due to increased demand for screens, but also an increased capacity buer. The increased buer results in lower average waiting time. The impact on total cost is non-monotonic which leads us to the joint optimization model where we nd through our numerical study that changes in the parameters which cause the optimal capacity to increase generally correspond to decreases in the optimal average time at home (and often the average wait time as well). An exception to this general pattern is the special case where patients only leave the system by getting cancer. In this case increasing the cancer rate would lead to lower capacity and lower average time at home as the provider wants to detect cancer faster, but patients also leave the system sooner. In reality, patients are much more likely to die from other causes than colorectal cancer, so this exception seems unlikely in practice.
From our analysis, we obtain several important insights for policy makers and cancer screen- ing providers. Our ndings suggest that if providers have the exibility to adjust capacity at fairly constant marginal cost, increased demand will lead to improved service delivery in terms
of reduced wait time for colonoscopy. We nd that as the new patient arrival rate increases, e.g. due to shifts in the population distribution or increased insurance coverage, the optimal capacity increases due to an increased demand for screens but also an increased capacity buer. If the new patient arrival rate increases approximately 400% at an average clinic, we see that the capacity increases approximately 399%. This is because the clinic is operating at very high utilization and therefore keeps a relatively low slack (capacity - input rate) compared to capacity itself. The increase in the capacity buer leads to lower average wait times. If clinics are capac- ity constrained, and choose only the average time at home, we see from our numerical study that increasing the new patient arrival rate by 50% leads to a roughly 50% increase in the average time spent at home with little eect on wait time. The social planner prefers that patients screen less frequently (wait longer on average until scheduling the next screen) when they are capacity constrained. This lengthens the eective screening guidelines to what the provider can reasonably deliver to the population. In reality, another way the shortage of capacity may play out is through increased wait-times for appointments rather than by adjustments in the time until the patient tries to schedule her next screen. Some patients may become frustrated with how lengthy appointment delays make it more challenging to adhere to guidelines and those patients may become less likely to be compliant. When social planners have the exibility to adjust provider capacity and inuence the average time patients spend at home, increasing the new patient arrival rate will again lead to lower wait times with little impact on the average time at home. This result is supported by the range of parameters estimated for our numerical study from public health data, and also derived analytically for a special case of the parameters. These results indicate that while growing demand is a concern in capacity-constrained environ- ments, it can also result in improved appointment access when resources are available to adjust
capacity accordingly.
A challenge with cancer screening, and colorectal cancer in particular, is asymptomatic disease progression and the accuracy of aordable alternatives for screening and diagnosis such as FOBT. In our model we nd that if the time until detection from other methods decreases, i.e. patients are encouraged to use FOBT tests with greater frequency and improved accuracy, this will reduce the optimal colonoscopy screening capacity required and likely decrease the optimal screening cost. Though the optimal cost is not generally monotone in the time at home, we observe it to only be increasing for a range of parameters which is unlikely in practice (e.g. average time until detection from other methods exceeds average lifetime). Lower screening capacity will lead to higher expected wait time for those who do request screens. In our numerical study of the joint optimization model, we also see that decreasing the time until detection via other methods will result in lower optimal capacity, higher optimal average waiting times, higher optimal average time at home, and lower optimal average cost. The provider prefers that patients wait at home longer so that the disease is more likely to be detected via other methods before requesting an expensive screen. An exception to these ndings in practice is that colonoscopy is often used as a follow-up screen to conrm suspected diagnosis from other methods (e.g. positive FOBT test or symptoms). The assumptions of our model could be adjusted to account for such a setting. It could be in that case that faster detection from other methods implies an increase in capacity along with higher wait times and either higher or lower costs. Another caveat is how these results should be interpreted across dierent types of other detection. For example, if cancer is being detected faster because it is spreading to other organs and parts of the body faster, then a correlation between lower time until detection from other causes and higher detection delay penalty for other detection may exist. Future research could examine the
eect of such a correlation or model cancer detected via other screens (e.g. FOBT) separately from cancer detected via late stage progression, symptoms, or death.
Using public health data collected for our numerical study, our analysis also provides esti- mates of current capacity needs, economic validation of guidelines, and service delivery perfor- mance measures. Assuming the average time at home to be xed at the current guideline of 10 years (520 weeks), we use public health data to estimate our model parameters and estimate that the average clinic sees a market size of 10.662 new patients per week and should provide an optimal capacity of 32.487 screens per week at an estimated long-run average weekly cost of $129,766 (capacity operating cost + delay cost). The average wait time will be 9.069 weeks. In order to guarantee that no more than 5% of patients are delayed more than 2 weeks for screening, the provider should choose a weekly capacity of 33.874 which results in an increase in the long-run average weekly costs of of 3.05% over optimal. While these estimates provide some insight into screening capacity needs, the model we develop is a stylized optimization model which was built for theoretical analysis as opposed to empirical estimation. Certain parameters of our model, such as the delay costs, are dicult to estimate due to the more implicit nature of what they represent. Initializing the model with data which is more specic to a particular market or geographic region would increase the reliability of the model estimates. We leave for future research further empirical estimation of colonoscopy screening capacity needs.
There are several opportunities for future research stemming from our model. One primary limitation of the model is that the exponential distribution assumptions may not fully charac- terize consumer behavior. The time patients spend at home might be more correctly modeled as a deterministic time (e.g. 10 years) plus some random time until the patient remembers to call and make the appointment or is reminded to do so by a primary care provider. We leave
this issue for future research, because while it may accurately describe the real-world behavior, the added complexity to the model may have little marginal benet in terms of the research questions posed in this paper. This paper primarily focuses on issues regarding the design of the queueing network; whereas future research may emphasize problems regarding queueing control such as how to prioritize screens based on risk factors and how long it has been since the patient was last screened. Other future research might investigate reasons for poor adherence rates for colonoscopy screening and consider to what degree poor operational issues such as appointment access are drivers for poor patient adherence to screening guidelines. Other work might use location network analysis to investigate issues relevant to state and federal policy makers such as where to invest in additional cancer screening resources and where to build new endoscopy suites. Finally, future research might extend the general approach used within this work to other types of cancer or other diseases. Some of the assumptions of this paper, and most of the parameter estimates for the numerical study, were specically motivated by the application to colorectal cancer; however the model is generalizable to screening for other diseases. Models which investigate the tradeos between supply planning and provider-driven demand in health services are beginning to be investigated more by the operations management and operations research community, and future work could bring about greater understanding of these types of problems.
Chapter 4
Revenue Management for Outpatient
Appointments
4.1 Introduction
Outpatient clinics use appointment systems to match supply and demand while balancing goals of provider service eciency with timely access for patients. Appointment scheduling helps to smooth work ow for resources and reduce variability in patient wait time for service. How- ever, poor appointment utilization and excessive delays for outpatient appointments are widely recognized as signicant barriers to eective health care delivery. Several studies document the high nancial costs associated with no-shows and cancellations as well as their impact on sta satisfaction and productivity. Moore et al. (2001) estimate the cost of no-shows and cancella- tions at 3% to 14% of revenues. Other studies emphasize the cost of long appointment delay, the time between the request for and service of an appointment, highlighting lower patient sat- isfaction, decreased quality of care, and system-wide costs incurred when patients end up in the emergency room for non-emergency treatment. Strunk and Cunningham (2002) conclude that the percent of patients reporting an inability to obtain a timely appointment rose from 23% to 33% from 1997 to 2001.
Figure 4.1: Attendance Rate by Appointment Delay
Much evidence suggests a relationship between appointment waste and appointment delay. The longer the appointment delay, the less likely the patient is to attend the appointment, and in turn, wasted appointments, due to no-shows and late cancellations, do nothing to reduce the backlog of patients waiting. Festinger et al. (2002) randomly assign outpatient clients to dierent appointment delays and nd that 72% of subjects scheduled 1 day later attend their appointments compared to 41% for 3 days later and 38% for 7 days later. Gallucci et al. (2005) nd a similar relationship between appointment delay and rate of kept appointments for referrals to an outpatient program at a community mental health center. Figure 4.1 demonstrates the relationship between appointment waste and delay across three types of clinics within a major university health care system. Clinics, therefore, have incentive to reserve capacity for patients who schedule closer to the appointment date as they more likely to attend the appointment. In this study, we analyze how clinics should allocate appointments to customer classes with hetero- geneous no-show rates by developing a capacity control and overbooking model for outpatient appointments.
success in practice because they emphasize either appointment waste or appointment delay without considering the relationship between them. Solutions which emphasize service eciency include no-show penalties, appointment reminders, and overbooking. In practice, it is dicult to collect nancial penalties from patients who do not attend appointments. Other solutions emphasize improving customer access by dedicating capacity or resources to urgent requests. Clinics may dedicate one resource to urgent requests such as a same-day doctor, and larger health care systems may administer urgent care clinics or minute clinics. Healthcare providers may also carve-out (reserve) capacity for urgent requests. Problems with these solutions are high xed costs associated with operating a resource dedicated to urgent problems and the resource costs associated with triaging which patients qualify to use capacity reserved for urgent requests. Clinics, such as private boutique clinics, may also reduce delay by limiting panel size, or new patients, as a means of controlling demand upfront.
Open access, also known as same-day scheduling or advanced access, is a popular paradigm in which the clinic attempts to do today's work today by oering each patient a same-day appointment and/or encouraging patients to book within a short window (Murray and Tantau 2000, Murray and Berwick 2003). Open access reduces the need for triage, by emphasizing patient-centered care as a means for simultaneously achieving lower appointment waste and delay simultaneously. Open access alleviates some problems, but if not managed eectively, can lead to greater variation in daily demand and conicts with patient preferences. A pure open access system, similar to a walk-in clinic where no advance appointments are allowed, could make the clinic more vulnerable to costly under- or over-utilization because of the absence of an appointment system buer. These costs can be particularly high in clinics with providers who are not in clinic full-time, such as academic medical centers with residents and faculty. Open
access may also conict with preferences of patients who must make advance arrangements such as transportation, absence from work, childcare, or lodging. The underlying issue of managing clinic capacity is not eliminated with open access. Clinics still must decide how to allocate appointments to dierent demand classes in a way that is both patient-centered and ecient. While open access is generally presented as an alternative to traditional advance scheduling, it is unclear exactly how the two may be eectively used together as a hybrid system.
How much xed service capacity to allocate to dierent demand classes and whether to over- book are problems faced in outpatient appointment scheduling, but they are common to many service industries. Within operations and management sciences, these problems are typically re- ferred to as capacity control decisions which fall under the broader eld of revenue management. When there are advance purchase and reservation options, rms must also decide whether to overbook their xed capacity to buer against no-shows and late cancellations. Demand classes are customarily dierentiated by features such as price and service package oered; however, dierences in no-show and cancellation rates among demand classes are particularly signicant in outpatient clinics which are limited in their ability to collect upfront deposits for service or to re-sell a wasted reservation to a late-arriving customer. Frequently, capacity allocation and overbooking decisions are considered separately, but several joint capacity-control and over- booking models, developed under the assumption that no-show probabilities are the same for all customers, show rms can achieve better results by integrating the two decisions (Talluri and van Ryzin 2005). We relax the assumption that no-show rates are homogeneous across demand classes, and show how rms can improve performance by accounting for these dierences.
We construct a joint overbooking and capacity control model to study the optimal appoint- ment allocation and overbooking decisions for outpatient clinics when patient demand classes
dier with respect to no-show rates. In Section 2, we review the literature on joint overbooking and capacity control and the literature on outpatient scheduling. In Section 3 we formulate a two-stage joint overbooking and capacity control model and provide structural results regarding the optimal policy. Under general assumptions, a closed form expression for the optimal pol- icy can be dicult to derive; therefore, we develop lower and upper bounds and near-optimal approximations. In Section 4 we conduct a numerical study to compare the performance of the optimal policy with bounds and approximations, as well as policies from previous literature and clinic practice. In Section 5 we provide conclusions and insights from our work.
4.2 Literature Review
Our work is positioned at the intersection of the revenue management literature addressing joint capacity control and overbooking and the outpatient appointment scheduling literature. Talluri and van Ryzin (2005) provide a comprehensive review of the revenue management literature. They consider capacity control and overbooking separately prior to integration. The simple two-class allocation rule proposed by Littlewood (1972), is a seminal concept in traditional capacity control models. Assuming both a discount and a full fare segment, with independent random demands where the full fare demand,D1, arrives later, Littlewood's rule species that
withx units of capacity remaining, a discount unit should be sold as long as the discount fare, p2, equals or exceeds the product of the full fare price, p1, and the probability that full fare